====== Block 16 - Ampère's Law and Magnetomotive Force (MMF) ====== ===== Learning objectives ===== After this 90-minute block, you can * ... ===== Preparation at Home ===== Well, again * read through the present chapter and write down anything you did not understand. * Also here, there are some clips for more clarification under 'Embedded resources' (check the text above/below, sometimes only part of the clip is interesting). For checking your understanding please do the following exercises: * ... ===== 90-minute plan ===== - Warm-up (x min): - .... - Core concepts & derivations (x min): - ... - Practice (x min): ... - Wrap-up (x min): Summary box; common pitfalls checklist. ===== Conceptual overview ===== - ... ===== Core content ===== ===== Generalization of the Magnetic Field Strength ===== So far, only the rotational symmetric problem of a single wire was considered in formula. I.e a current $I$ and the length $s$ of a magnetic field line around the wire was given to calculate the magnetic field strength $H$: \begin{align*} \quad H_\varphi ={I\over{s}} = {{I}\over{2 \cdot \pi \cdot r}} \quad \Leftrightarrow \quad I = H_\varphi \cdot {s} \quad \quad \quad | \quad \text{applies only to the long, straight conductor} \end{align*} Now, this shall be generalized. For this purpose, we will look back at the electric field. \\ For the electric field strength $E$ of a capacitor with two plates at a distance of $s$ and the potential difference $U$ holds: \begin{align*} U = E \cdot s \quad \quad | \quad \text{applies to plate capacitor only} \end{align*} In words: The potential difference is given by adding up the field strength along the path of a probe from one plate to the other. \\ This was extended to the vltage between to points $1$ and $2$. Additionally, we know by Kirchhoff's voltage law that the voltage on a closed path is "0". \begin{align*} U_{12} &= \int_1^2 \vec{E} \; {\rm d}\vec{s} \\ U &= \oint \vec{E} \; {\rm d}\vec{s} =0 \\ \end{align*} We can now try to look for similarities. Also for the magnetic field, the magnitude of the field strength is summed up along a path to arrive at another field-describing quantity. \\ Because of the similarity the so-called **magnetic potential difference $V_m$** between point $1$ and $2$ is introduced: \begin{align*} V_m &= H \cdot s \quad \quad | \quad \text{applies to rotational symmetric problems only} \\ \end{align*} \begin{align*} \boxed{ V_m = V_{m, 12} = \int_1^2 \vec{H} \; {\rm d}\vec{s} \\ V_m = \oint \vec{H} {\rm d}\vec{s} = \theta } \end{align*} We need to take a loser look here. Any closed path in the static electric field leads to a potential difference of $U = \oint \vec{E} \; {\rm d}\vec{s} =0$. \\ BUT: closed paths in the static magnetic field leads to a magnetic potential difference which is **not mandatorily** $0$! $ V_m = \oint \vec{H} {\rm d}\vec{s} = \theta$ \\ \\ Another new quantity is introduced: the **magnetic voltage $\theta$**: - The magnetic voltage $\theta$ is the magnetic potential difference on a closed path. - Since the magnetic voltage $\theta$ is valid for exactly __one turn__ along our single wire, $\theta$ is also equal to the current through the wire: \\ \begin{align*} \theta = H \cdot s = I \quad \quad | \quad \text{applies only to the long, straight conductor} \end{align*} - The magnetic potential difference can take a fraction or a multiple of one turn and is therefore **not mandatorily** equal to $I$. - The magnetic voltage is generalized in the following box. The path integral of the magnetic field strength along an arbitrary closed path is equal to the free currents (= current density) through the surface enclosed by the path. The magnetic voltage $\theta$ (and therefore the current) is the cause of the magnetic field strength. \\ This leads to the **{{wp>Ampere's Circuital Law}}** | \begin{align*} \boxed{\oint_{s} \vec{H} {\rm d} \vec{s} = \theta } \end{align*}| The magnetic voltage $\theta$ can be given as \\ (nbsp)(nbsp) • $\theta = I \quad \quad \quad \ $ for a single conductor \\ (nbsp)(nbsp) • $\theta = N \cdot I \quad \:\; \, $ for a coil\\ (nbsp)(nbsp) • $\theta = \sum_n \cdot I_n \quad$ for multiple conductors\\ (nbsp)(nbsp) • $\theta = \iint_A \; \vec{S} {\rm d}\vec{A}$ for any spatial distribution (see [[block15]])| The unit of the magnetic voltage $\theta$ is **Ampere** (or **Ampere-turns**). In the english literature the magnetic voltage is called **{{wp>Magnetomotive force}}** ${\rm d}\vec{s}$ and ${\rm d}\vec{A}$ in $\oint_{s} \vec{H} {\rm d} \vec{s} = \theta = \iint_A \; \vec{S} {\rm d}\vec{A}$ build a right-hand system. \\ - Once the thumb of the right hand is pointing along ${\rm d}\vec{A}$, the fingers of the right hand show the correct direction for ${\rm d}\vec{s}$ for positive $\vec{H}$ and $\vec{S}$ - Currents into the direction of the right hand's thumb count positive. Currents antiparallel to it count negative. \\ {{drawio>Righthandrule.svg}} ~~PAGEBREAK~~ ~~CLEARFIX~~ ==== Recap of the fieldline images ==== === longitudinal coil === \\ {{url>https://www.falstad.com/vector3dm/vector3dm.html?f=SolenoidField&d=streamlines&sl=none&st=3&ld=5&a1=21&a2=30&a3=100&rx=63&ry=1&rz=2&zm=2.396 700,450 noborder}} A longitudinal coil can be seen in . \\ The magnetic field in a toroidal coil is often considered as homogenious in the inner volume, when the length $l$ is much larger than the diameter: $l \gg d$. \\ With a given number $N$ of windings, the magnetic field strength $H$ is \begin{align*} \theta = H \cdot l = N \cdot I \end{align*} \begin{align*} \boxed{H = {{N \cdot I}\over{l}}} \biggr | _\text{longitudinal coil} \end{align*} === toroidal coil === \\ {{url>https://www.falstad.com/vector3dm/vector3dm.html?f=ToroidalSolenoidField&d=streamlines&sl=none&st=1&ld=8&a1=77&a2=26&a3=100&rx=0&ry=0&rz=0&zm=1.8 700,450 noborder}} A toroidal coil has a donut-like setup. This can be seen in . \\ The toroidal coil is often defined by: * The minor radius $r$: The radius of the circular cross-section of the coil. * The major radius $R$: The distance from the center of the entire toroid (the center of the hole) to the center of the circular cross-section of the coil. For reasons of symmetry, it shall get clear that the field lines form concentric circles. \\ Also the magnetic field strength $H$ in a toroidal coil is often considered as homogenious, when $R \gg r$. With a given number $N$ of windings, the magnetic field strength $H$ is \begin{align*} \theta = H \cdot 2\pi R = N \cdot I \end{align*} \begin{align*} \boxed{H = {{N \cdot I}\over{2\pi R}}} \biggr | _\text{toroidal coil} \end{align*} ===== Common pitfalls ===== * ... ===== Exercises ===== {{fa>pencil?32}} {{drawio>Task3MagneticFieldCurrentFlowingConductor.svg}} \\ Given are the adjacent closed trajectories in the magnetic field of current-carrying conductors (see ). Let $I_1 = 2~\rm A$ and $I_2 = 4.5~\rm A$ be valid. In each case, the magnetic potential difference $V_{\rm m}$ along the drawn path is sought. #@HiddenBegin_HTML~323100,Path~@# * The magnetic potential difference is given as the **sum of the current through the area within a closed path**. * The direction of the current and the path have to be considered with the righthand rule. #@HiddenEnd_HTML~323100,Path~@# #@HiddenBegin_HTML~323102,Result a)~@# a) $V_{\rm m,a} = - I_1 = - 2~\rm A$ \\ #@HiddenEnd_HTML~323102,Result~@# #@HiddenBegin_HTML~323103,Result b)~@# b) $V_{\rm m,b} = - I_2 = - 4.5~\rm A$ \\ #@HiddenEnd_HTML~323103,Result~@# #@HiddenBegin_HTML~323104,Result c)~@# c) $V_{\rm m,c} = 0 $ \\ #@HiddenEnd_HTML~323104,Result~@# #@HiddenBegin_HTML~323105,Result d)~@# d) $V_{\rm m,d} = + I_1 - I_2 = 2~\rm A - 4.5~\rm A = - 2.5~\rm A$ \\ #@HiddenEnd_HTML~323105,Result~@# #@HiddenBegin_HTML~323106,Result e)~@# e) $V_{\rm m,e} = + I_1 = + 2~\rm A$ \\ #@HiddenEnd_HTML~323106,Result~@# #@HiddenBegin_HTML~323107,Result f)~@# f) $V_{\rm m,f} = 2 \cdot (- I_1) = - 4~\rm A$ \\ #@HiddenEnd_HTML~323107,Result~@# {{page>electrical_engineering_and_electronics:task_jfzlmsucghsqvop5_with_calculation&nofooter}} {{page>electrical_engineering_and_electronics:task_kmp8r8y6lvwjnoc3_with_calculation&nofooter}} {{page>electrical_engineering_and_electronics:task_ddjurcpk494go2q1_with_calculation&nofooter}} ===== Embedded resources ===== Explanation (video): ... ~~PAGEBREAK~~ ~~CLEARFIX~~